Comparing Counts

Chapter 26

Goodness-of-Fit

• A test of whether the distribution of counts in one categorical variable matches the distribution predicted by a model is called a goodness-of-fit test.

• As usual, there are assumptions and conditions to consider…

Assumptions and Conditions

• Counted Data Condition: Check that the data are counts for the categories of a categorical variable.

• Independence Assumption: The counts in the cells should be independent of each other.– Randomization Condition: The individuals who

have been counted and whose counts are available for analysis should be a random sample from some population.

Assumptions and Conditions

• Sample Size Assumption: We must have enough data for the methods to work.– Expected Cell Frequency Condition: We should

expect to see at least 5 individuals in each cell.• This is similar to the condition that np and nq

be at least 10 when we tested proportions.

Calculations

• Since we want to examine how well the observed data reflect what would be expected, it is natural to look at the differences between the observed and expected counts (Obs – Exp).

• These differences are actually residuals, so we know that adding all of the differences will result in a sum of 0. That’s not very helpful.

Calculations

• We’ll handle the residuals as we did in regression, by squaring them.

• To get an idea of the relative sizes of the differences, we will divide each squared difference by the expected count fro that cell.

Calculations

• The test statistic, called the chi-square (or chi-squared) statistic, is found by adding up the sum of the squares of the deviations between the observed and expected counts divided by the expected counts:

22

all cells

Obs Exp

Exp

Calculations

• The chi-square models are actually a family of distributions indexed by degrees of freedom (much like the t-distribution).

• The number of degrees of freedom for a goodness-of-fit test is n – 1, where n is the number of categories.

One-Sided or Two-Sided?

• The chi-square statistic is used only for testing hypotheses, not for constructing confidence intervals.

• If the observed counts don’t match the expected, the statistic will be large—it can’t be “too small.”

• So the chi-square test is always one-sided.– If the calculated statistic value is large enough,

we’ll reject the null hypothesis.

One-Sided or Two-Sided?

• The mechanics may work like a one-sided test, but the interpretation of a chi-square test is in some ways many-sided.

• There are many ways the null hypothesis could be wrong.

• There’s no direction to the rejection of the null model—all we know is that it doesn’t fit.

The Chi-Square Calculation

1. Find the expected values: – Every model gives a hypothesized proportion for

each cell. – The expected value is the product of the total

number of observations times this proportion.2. Compute the residuals: Once you have

expected values for each cell, find the residuals, Observed – Expected.

3. Square the residuals.

The Chi-Square Calculation

4. Compute the components. Now find the components

for each cell.5. Find the sum of the components (that’s the

chi-square statistic).

2Observed Expected

Expected

The Chi-Square Calculation

6. Find the degrees of freedom. It’s equal to the number of cells minus one.

7. Test the hypothesis. Use your chi-square statistic to find the P-value.

(Remember, you’ll always have a one-sided test.) Large chi-square values mean lots of deviation

from the hypothesized model, so they give small P-values.

But I Believe the Model…

• Goodness-of-fit tests are likely to be performed by people who have a theory of what the proportions should be, and who believe their theory to be true.

• Unfortunately, the only null hypothesis available for a goodness-of-fit test is that the theory is true.

• As we know, the hypothesis testing procedure allows us only to reject or fail to reject the null.

But I Believe the Model…

• We can never confirm that a theory is in fact true.

• At best, we can point out only that the data are consistent with the proposed theory.– Remember, it’s that idea of “not guilty” versus

“innocent.”

Comparing Observed Distributions

• We can never confirm that a theory is in fact true. A test comparing the distribution of counts for two or more groups on the same categorical variable is called a chi-square test of homogeneity.

• A test of homogeneity is actually the generalization of the two-proportion z-test.

Comparing Observed Distributions

• The statistic that we calculate for this test is identical to the chi-square statistic for goodness-of-fit.

• In this test, however, we ask whether choices are the same among different groups (i.e., there is no model).

• The expected counts are found directly from the data and we have different degrees of freedom.

Assumptions and Conditions

• The assumptions and conditions are the same as for the chi-square goodness-of-fit test:– Counted Data Condition: The data must be counts.– Randomization Condition and 10% Condition: As

long as we don’t want to generalize, we don’t have to check these conditions.

– Expected Cell Frequency Condition: The expected count in each cell must be at least 5.

Calculations

• To find the expected counts, we multiply the row total by the column total and divide by the grand total.

• We calculated the chi-square statistic as we did in the goodness-of-fit test:

• In this situation we have (R – 1)(C – 1) degrees of freedom, where R is the number of rows and C is the number of columns.– We’ll need the degrees of freedom to find a

P-value for the chi-square statistic.

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all cells

Obs Exp

Exp

Chi Squared

• GOF: one categorical variable matches the distribution predicted by the model

• Homogeneity: A test comparing the distribution of counts for two or more groups on the same categorical variable.

• Independence: A test of whether two categorical variables are independent examines the distribution of counts for one group of individuals classified according to both variables.

Examining the Residuals

• When we reject the null hypothesis, it’s always a good idea to examine residuals.

• For chi-square tests, we want to work with standardized residuals, since we want to compare residuals for cells that may have very different counts.

• To standardize a cell’s residual, we just divide by the square root of its expected value:

Obs Expc

Exp

Examining the Residuals

• These standardized residuals are just the square roots of the components we calculated for each cell, with the + or the – sign indicating whether we observed more cases than we expected, or fewer.

• The standardized residuals give us a chance to think about the underlying patterns and to consider the ways in which the distribution might not match what we hypothesized to be true.

Independence

Contingency tables categorize counts on two (or more) variables so that we can see whether the distribution of counts on one variable is contingent on the other.

A test of whether the two categorical variables are independent examines the distribution of counts for one group of individuals classified

A chi-square test of independence uses the same calculation as a test of homogeneity

Assumptions and Conditions

• We still need counts and enough data so that the expected values are at least 5 in each cell.

• If we’re interested in the independence of variables, we usually want to generalize from the data to some population.– In that case, we’ll need to check that the data are

a representative random sample from that population.

Examine the Residuals

• Each cell of a contingency table contributes a term to the chi-square sum.

• It helps to examine the standardized residuals, just like we did for tests of homogeneity.

Chi-Square and Causation

• Chi-square tests are common, and tests for independence are especially widespread.

• We need to remember that a small P-value is not proof of causation.– Since the chi-square test for independence treats

the two variables symmetrically, we cannot differentiate the direction of any possible causation even if it existed.

– And, there’s never any way to eliminate the possibility that a lurking variable is responsible for the lack of independence.

Chi-Square and Causation

• In some ways, a failure of independence between two categorical variables is less impressive than a strong, consistent, linear association between quantitative variables.

• Two categorical variables can fail the test of independence in many ways.– Examining the standardized residuals can help you

think about the underlying patterns.

What Can Go Wrong?• Don’t use chi-square methods unless you have counts.– Just because numbers are in a two-way table doesn’t

make them suitable for chi-square analysis.• Beware large samples.– With a sufficiently large sample size, a chi-square test

can always reject the null hypothesis.• Don’t say that one variable “depends” on the other just

because they’re not independent.– Association is not causation.

What have we learned?

• We’ve learned how to test hypotheses about categorical variables.

• All three methods we examined look at counts of data in categories and rely on chi-square models.– Goodness-of-fit tests compare the observed distribution of a

single categorical variable to an expected distribution based on theory or model.

– Tests of homogeneity compare the distribution of several groups for the same categorical variable.

– Tests of independence examine counts from a single group for evidence of an association between two categorical variables.